Remark

The theorem extends verbatim to finite $Z$, with one correction $s_i$ per point. For XV-D's example ($X={\mathbb{R}}^2$, $Z=\{0\}$, $T^{\circ }=\delta (x_1){|}_{X^{\circ }}$, $\tilde{T}=\delta (x_1)+{\partial }_1{\delta }_0$, $k=1$) one finds, for $g\in A_B$, $\lambda (g)=\int g(0,x_2)\mathrm{d}{x}_2-{\partial }_{1}g(0)=-{\partial }_{1}g(0)$, so $\Lambda (j)=-j_{e_1}$, $c_{e_1}=-1$, $s={\partial }_1{\delta }_0$, and $T=\delta (x_1)$, in agreement with XVD.